Contents

Idea

The Moore complex of a simplicial group – also known in its normalized version as the complex of normalized chains – is a chain complex whose differential is built from the face maps of the simplicial group.

Definition

Given a simplicial groupAA (or in fact any simplicial set), then an element a∈An+1a \in A_{n+1} is called degenerate (or thin) if it is in the image of one of the simplicial degeneracy maps si:An→An+1s_i \colon A_n \to A_{n+1}. All elements of A0A_0 are regarded a non-degenerate. Write

Properties

Normalization

Proposition

from the normalized chain complex, def. 1, into the alternating face map complex modulo degeneracies, def. 4, (inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.

Hypercrossed complex structure

Proposition

This has been established in (Carrasco-Cegarra). In fact, the analysis of the Moore complex and what is necessary to rebuild the simplicial group from its Moore complex is the origin of the abstract motion of hypercrossed complex, so our stated proposition is almost a tautology!

Suppose that GG is a simplicial group with Moore complex NGN G, which satisfies NGk=1N G_k = 1 for k>1k\gt 1, then (G1,G0,d1,d0)(G_1,G_0,d_1,d_0) has the structure of a 2-group. The interchange law is satisfied since the corresponding equation in G1G_1 is always the image of an element in NG2N G_2, and here that must be trivial. If one thinks of the 2-group as being specified by a crossed module(C,P,δ,a)(C,P,\delta, a), then in terms of the original simplicial group, GG, NG0=G0=PN G_0 = G_0 = P, NG1≅CN G_1 \cong C, ∂=δ \partial = \delta and the action of PP on CC translates to an action of NG0N G_0 on NG1N G_1 using conjugation by s0(p)s_0(p), i.e., for p∈G0p\in G_0 and c∈NG1c\in N G_1,

a(p)(c)=s0(p)cs0(p)−1.a(p)(c) = s_0(p)c s_0(p)^{-1}.

Suppose next that NGk=1N G_k = 1 for k>2k \gt 2, then the Moore complex is a 2-crossed module.

(where we are indicating only the face maps for notational simplicity).

Here the first ℤ2=ℤ⊕ℤ\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}, the direct sum of two copies of the integers, is the group of 0-chains generated from the two endpoints (0)(0) and (1)(1) of Δ[1]\Delta[1], i.e. the abelian group of formal linear combinations of the form

The second ℤ3≃ℤ⊕ℤ⊕ℤ\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z} is the abelian group generated from the three (!) 1-simplicies in Δ[1]\Delta[1], namely the non-degenerate edge (0→1)(0\to 1) and the two degenerate cells (0→0)(0 \to 0) and (1→1)(1 \to 1), hence the abelian group of formal linear combinations of the form

Now of course most of the (infinitely!) many simplices inside Δ[1]\Delta[1] are degenerate. In fact the only non-degenerate simplices are the two 0-cells (0)(0) and (1)(1) and the 1-cell (0→1)(0 \to 1). Hence the alternating face maps complex modulo degeneracies, def. 4, of ℤ[Δ[1]]\mathbb{Z}[\Delta[1]] is simply this:

Notice that alternatively we could consider the topological 1-simplex Δ1=[0,1]\Delta^1 = [0,1] and its singular simplicial complexSing(Δ1)Sing(\Delta^1) in place of the smaller Δ[1]\Delta[1], then the free simplicial abelian group ℤ(Sing(Δ1))\mathbb{Z}(Sing(\Delta^1)) of that. The corresponding alternating face map chain complex C(ℤ(Sing(Δ1)))C(\mathbb{Z}(Sing(\Delta^1))) is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular nn-simplex in [0,1][0,1] is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.

Notice that these authors write “normalized chain complex” for the complex that elsewhere in the literature would be called just “Moore complex”, whereas what Goerss–Jardine call “Moore complex” is sometime maybe just called “alternating sum complex”.

A discussion with an emphasis of the generalization to non-abelian simplicial groups is in section 1.3.3 of